If \(x=a \sin ^2 \theta, y=a \cos ^2 \theta\) then \(\frac{d y}{d x}\) =

The maximum value of \(4 \sin ^2 x+3 \cos ^2 x\) is:

Area (in sq. units) of the region bounded by the curve \(y^2=4 x, y\)-axis and the line \(y=3\) is

Let \(\vec{a}\) and \(\vec{b}\) be unit vectors. If \(\sqrt{3} \vec{a}-\vec{b}\) is a unit vector, then the angle between \(\vec{a}\) and \(\vec{b}\) is

A box contains 10 balls, each marked with one of the digits 0 to 9 .
If four balls are drawn successively with replacement from the box,
then the probability that none is marked with the digit \(0\) is :

If \(A\) is any event associated with sample space and If \(E_1, E_2, E_3\) are mutually exclusive and exhaustive events. Then which of the following are true?
(A) \(P(A)=P\left(E_1\right) P\left(E_1 \mid A\right)+P\left(E_2\right) P\left(E_2 \mid A\right)+P\left(E_3\right) P\left(E_3 \mid A\right)\)
(B) \(P(A)=P\left(A \mid E_1\right) P\left(E_1\right)+P\left(A \mid E_2\right) P\left(E_2\right)+P\left(A \mid E_3\right) P\left(E_3\right)\)
(C) \(P\left(E_i \mid A\right)=\frac{P\left(A \mid E_i\right) P\left(E_i\right)}{\sum_{i=1}^3 P\left(A \mid E_i\right) P\left(E_i\right)}, i=1,2,3\)
(D) \(P\left(A \mid E_i\right)=\frac{P\left(E_i \mid A\right) P\left(E_i\right)}{\sum_{i=1}^3 P\left(E_i \mid A\right) P\left(E_i\right)}, i=1,2,3\)
Choose the correct answer from the options given below:

For a manufacturer, total cost function is given by \(C=\frac{x^2}{25}+2 x\). Which of the following are correct ?
A. 2.6 is the marginal cost when 5 units are produced.
B. \(\frac{2 x}{25}+2\) is the marginal cost function.
C. \(\frac{x}{25}+2\) is the average cost function.
D. 2.4 is the marginal cost when 5 units are produced.
E. \(\frac{x}{25}+1\) is the average cost function.
Choose the correct answer from the options given below:

\(\text{[NiCl}_2\text{(PPh}_3)_2\text{]}\) is named as :

A negatively charged object A is repelled by another charged object B. However, an object C is attracted to object B.
Which of the following is the most probable for the charge of object C?

Let the vectors \(\vec{a}=\hat{i}-3 \hat{j}+2 \hat{k}, \vec{b}=2 \hat{i}+\hat{j}-\hat{k}\) and \(\overrightarrow{ c }=3 \hat{ i }+5 \hat{ j }-2 \lambda \hat{ k }\) be coplanar. Then \(\lambda\) is equal to

The self-inductance of an inductance coil having 100 turns is 20 mH. The total flux through the coil for a current of 40 mA is :

Which of the following will NOT be coloured?

Match List I with List II and choose the correct answer from the options given below:

List - I	List - II
(A) \(\int \frac{\sin x}{1+\cos x} d x\)	(I) \(e^{\tan ^{-1} z}+C\)
(B) \(\int \frac{1}{1-\tan x} d x\)	(II) \(\log (\log x+1)+C\)
(C) \(\int \frac{e^{\tan ^{-1} x}}{1+x^2} d x\)	(III) \(-\log |1+\cos x|+C\)
(D) \(\int \frac{1}{x+x \log x} d x\)	(IV) \(\frac{x}{2}-\frac{1}{2} \log |\cos x-\sin x|+C\)